I recently came upon a recursive formula for the (ordinary) signatures of torus knots. The formula, which I found in Murasugi's book "Knot Theory and Applications" (Springer, 2007), originally appeared in a paper by Gordon, Litherland and Murasugi, "Signatures of Covering Links" (http://www.maths.ed.ac.uk/~aar/papers/glm.pdf). It is apparent from the formula that torus knots have even signatures. I wondered if this is a property specific to torus knots, so I went and poked around the Knot Atlas for a while. I did not see any knot of odd signature. (I did not check every knot, but checked enough to get discouraged.)

My question is: does there exist a knot of odd signature?

Secondly, why do all torus knots (and, quite possibly, all knots) have even signature?

isalways even? – zygund May 13 at 23:11